| L(s) = 1 | + 2.92·3-s − 2.54·5-s − 2.70·7-s + 5.57·9-s − 2.51·11-s + 1.02·13-s − 7.45·15-s − 0.352·17-s + 7.25·19-s − 7.91·21-s − 2.93·23-s + 1.48·25-s + 7.55·27-s + 1.15·29-s − 3.02·31-s − 7.35·33-s + 6.88·35-s − 8.69·37-s + 2.98·39-s + 1.09·41-s − 4.12·43-s − 14.2·45-s + 2.15·47-s + 0.308·49-s − 1.03·51-s − 2.66·53-s + 6.39·55-s + ⋯ |
| L(s) = 1 | + 1.69·3-s − 1.13·5-s − 1.02·7-s + 1.85·9-s − 0.756·11-s + 0.283·13-s − 1.92·15-s − 0.0855·17-s + 1.66·19-s − 1.72·21-s − 0.611·23-s + 0.297·25-s + 1.45·27-s + 0.214·29-s − 0.542·31-s − 1.28·33-s + 1.16·35-s − 1.43·37-s + 0.478·39-s + 0.170·41-s − 0.629·43-s − 2.11·45-s + 0.314·47-s + 0.0440·49-s − 0.144·51-s − 0.365·53-s + 0.862·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 269 | \( 1 + T \) |
| good | 3 | \( 1 - 2.92T + 3T^{2} \) |
| 5 | \( 1 + 2.54T + 5T^{2} \) |
| 7 | \( 1 + 2.70T + 7T^{2} \) |
| 11 | \( 1 + 2.51T + 11T^{2} \) |
| 13 | \( 1 - 1.02T + 13T^{2} \) |
| 17 | \( 1 + 0.352T + 17T^{2} \) |
| 19 | \( 1 - 7.25T + 19T^{2} \) |
| 23 | \( 1 + 2.93T + 23T^{2} \) |
| 29 | \( 1 - 1.15T + 29T^{2} \) |
| 31 | \( 1 + 3.02T + 31T^{2} \) |
| 37 | \( 1 + 8.69T + 37T^{2} \) |
| 41 | \( 1 - 1.09T + 41T^{2} \) |
| 43 | \( 1 + 4.12T + 43T^{2} \) |
| 47 | \( 1 - 2.15T + 47T^{2} \) |
| 53 | \( 1 + 2.66T + 53T^{2} \) |
| 59 | \( 1 + 7.47T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 6.89T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 - 0.0902T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 7.80T + 83T^{2} \) |
| 89 | \( 1 - 0.397T + 89T^{2} \) |
| 97 | \( 1 - 1.05T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.945198322818822834328045353082, −7.53928786585043002108823535383, −6.95456727924747600338435430983, −5.85593420328570223968053493212, −4.76406254002509312902180381018, −3.84177057707705627819759717268, −3.28051069539121598667366424038, −2.85603906964961265469184350587, −1.60894085843803229738741102040, 0,
1.60894085843803229738741102040, 2.85603906964961265469184350587, 3.28051069539121598667366424038, 3.84177057707705627819759717268, 4.76406254002509312902180381018, 5.85593420328570223968053493212, 6.95456727924747600338435430983, 7.53928786585043002108823535383, 7.945198322818822834328045353082
